# Substructural logic

In mathematical logic, in particular in connection with proof theory, a number of**substructural logics**have been introduced, as systems of propositional calculus that are weaker than the conventional one. They differ in having fewer

**structural rules**available: the concept of structural rule is based on the sequent presentation, rather than the natural deduction formulation. Two of the more significant substructural logics are relevant logic and linear logic.

In discussing the sequent calculus, one writes each line of a proof as

- .

- (
*A***and***B*)**implies***C*.

*C*(which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol.

Since conjunction is a commutative and associative operation, the formal setting-up of sequent theory normally includes **structural rules** for rewriting the sequent Γ accordingly - for example for deducing

- .

*idempotent*and

*monotonic*properties of conjunction: from

- .

*B*,

- .

*B*is clearly irrelevant to the conclusion.

These are basic examples of structural rules. It is not that these rules are contentious, when applied in conventional propositional calculus. They occur naturally in proof theory, and were first noticed there (before receiving a name).